On the $\tau$-compactness of products of $\tau$-measurable operators adjoint to semi-finite von Neumann algebras

Let ${\mathcal M}$ be the von Neumann algebra of operators in a Hilbert space $\mathcal H$ and $\tau$ be an exact normal semi-finite trace on $\mathcal{M}$. We obtain inequalities for permutations of products of $\tau$-measurable operators. We apply these inequalities to obtain new submajorizations (in the sense of Hardy, Littlewood, and Pólya) of products of $\tau$-measurable operators and a sufficient condition of orthogonality of certain nonnegative $\tau$-measurable operators. We state sufficient conditions of the $\tau$-compactness of products of self-adjoint $\tau$-measurable operators and obtain a criterion of the $\tau$-compactness of the product of a nonnegative $\tau$-measurable operator and an arbitrary $\tau$-measurable operator. We present an example that shows that the nonnegativity of one of factors is substantial. We also state a criterion of the elementary nature of the product of nonnegative operators from $\mathcal{M}$. All results are new for the *-algebra $\mathcal{B}(\mathcal{H})$ of all bounded linear operators in $\mathcal{H}$ endowed with the canonical trace $\tau=\operatorname{tr}$.